Information and Communication Technology Seminar, Vol. 1 No. 1, August 2005
ISSN 1858-1633 2005 ICTS 1
MATHEMATICAL MORPHOLOGY AND ITS APPLICATIONS
Akira Asano
Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University
Kagamiyama 1-7-1, Higashi-Hiroshima, Hiroshima 739-8521, JAPAN email: asanomis.hiroshima-u.ac.jp
ABSTRACT
This invited talk presents the concept of mathematical morphology, which is a mathematical
framework of quantitative image manipulations. The basic operations of mathematical morphology, the
relationship to image processing filters, the idea of size distribution and its application to texture analysis
are explained.
1. INTRODUCTION
Mathematical morphology treats an effect on an image as an effect on the shape and size of objects
contained in the image. Mathematical morphology is a mathematical system to handle such effects
quantitatively based on set operations [1–5].The word stem “morpho-” originates in a Greek word meaning
“shape,” and it appears in the word “morphing,” which is a technique of modifying an image into
another image smoothly.
The founders of mathematical morphology, G. Math´eron and J. Serra, were researchers of l ´ Ecole
Nationale Sup´erieure des Mines de Paris in France, and had an idea of mathematical morphology as a
method of evaluating geometrical characteristics of minerals in ores [6]. Math´eron is also the founder of
the random closed set theory, which is a fundamental theory of treating random shapes, and kriging, which
is a statistical method of estimating a spatial distribution of mineral deposits from trial diggings.
Mathematical morphology has relationships to these theories and has been developed as a theoretical
framework of treating spatial shapes and sizes of objects. The International Symposium on
Mathematical Morphology ISMM, which is the topical international symposium focusing on
mathematical morphology only, has been organized almost every two years, and its seventh symposium
was held in April 2005 in Paris as a cerebration of 40 years anniversary of mathematical morphology [7].
The paper explains the framework of mathematical morphology, especially opening, which is the
fundamental operation of describing operations on shapes and sizes of objects quantitatively, in Sec. 2.
Section 3. proves “filter theorem,” which guarantees that all practical image processing filters can be
constructed by combinations of morphological operations. Examples of expressing median filters and
average filters by combinations of morphological operations are also shown in this section. Section 4.
explains granulometry, which is a method of measuring the distribution of sizes of objects in an
image, and shows an application to texture analysis by the author.
2. BASIC OPERATIONS OF MATHEMATICAL MORPHOLOGY
The fundamental operation of mathematical morphology is “opening,” which discriminates and
extracts object shapes with respect to the size of objects. We explain opening on binary images at first,
and basic operations to describe opening.
2.1 Opening
In the context of mathematical morphology, an object in a binary image is regarded as a set of vectors
corresponding to the points composing the object. In the case of usual digital images, a binary image is
expressed as a set of white pixels or pixels of value one. Another image set expressing an effect to the
above image set is considered, and called structuring element. The structuring element corresponds to the
window of an image processing filter, and is considered to be much smaller than the target image to
be processed.
Let the target image set be X, and the structuring element be B. Opening of X by B has a property as
follows:
where Bz indicates the translation of B by z, defined as follows:
This property indicates that the opening of X with respect to B indicates the locus of B itself sweeping all
the interior of X, and removes smaller white regions than the structuring element, as illustrated in Fig. 1.
Since opening eliminates smaller structures and smaller bright peaks than the structuring element, it
has a quantitative smoothing ability.
Information and Communication Technology Seminar, Vol. 1 No. 1, August 2005
ISSN 1858-1633 2005 ICTS 2
Fig. 1. Effect of opening.
2.2. Fundamental Operations of Mathematical Morphology
Although the property of opening in E. 1 is intuitively understandable, this is not a pixelwise
operation. Thus opening is defined by a composition of simpler pixelwise operations. In order to define
opening, Minkowski set subtraction and addition are defined as the fundamental operations of mathematical
morphology.
Minkowski set subtraction has the following property: It follows from x
∈ X
b
that x − b
∈ X. Thus the definition of Minkowski set subtraction in Eq. 3
can be rewritten to the following pixelwise operation:
The reflection of B, denoted ˇB , is defined as
follows:
Using the above expressions, Minkowski set subtraction is expressed as follows:
Since we get from the definition of reflection in Eq. 6 that
, it follows that
. We get the relationship in Eq. 7 by substituting it into Eq.
5.This relationship indicates that is the
locus of the origin of when
sweeps all the interior of X.
For Minkowski set addition, it follows that Thus we get
Fig. 2. opening composed of fundamental operations.
It indicates that is composed by pasting a
copy of B at every point within X. Using the above operations, erosion and dilation
of X with respect to B are defined as and
. respectively. We get from Eq. 7 that
It indicates that is the locus of the origin
of B when B sweeps all the interior of X. The opening X
B
is then defined using the above fundamental operations as follows:
The above definition of opening is illustrated in Fig. 2. A black dot indicates a pixel composing an
image object in this figure. As shown in the above, the erosion of X by B is the locus of the origin of B when
B sweeps all the inside of X. Thus the erosion in the first step of opening produces every point where a
copy of B included in X can be located. The Minkowski addition in the second step locates a copy
of B at every point within
. Thus the opening of X with respect to B indicates the locus of B itself
sweeping all the interior of X, as described at the beginning of this section. In other words, the opening
removes regions of X which are too small to include a copy of B and preserves the others.
The counterpart of opening is called closing, defined as follows:
The closing of X with respect to B is equivalent to the opening of the background, and removes smaller
spots than the structuring element within image objects. This is because the following relationship
between opening and closing holds:
where X
c
indicate the complement of X and defined as
The relationship of Eq. 13 is called duality of opening and closing
1
.
1
There is another notation system which denotes opening as X
◦ B and closing as X • B.
Mathematical Morphology and Its Application – Akira Asano
ISSN 1858-1633 2005 ICTS 3
Figure 3 summarizes the illustration of the effects of basic morphological operations
2
.
Fig. 3. Effects of erosion, dilation, opening, and closing
Fig. 4. Umbra. The spatial axis x is illustrated one-dimensional for simplicity.
2.3. In the Case of Gray Scale Images
